1. 2nd INTERNATIONAL Conference on Nuclear and Renewable Energy Resources
4-7 July 2010, Gazi University, Ankara, Turkey
The Plasma Focus- Trending into the Future
Sing Lee 1,2,3* and Sor Heoh Saw 1,2
1INTI International University, Nilai, Malaysia
2Institute for Plasma Focus Studies, 32 Oakpark Drive, Chadstone, VIC 3148, Australia
3Nanyang Technological University, National Institute of Education, Singapore s:

2. Contents Introduction- Neutron scaling laws with energy
Scaling properties of the Plasma Focus
The Lee model code
Neutron scaling law deterioration and saturation-the cause
Connecting the behaviour of the scaling laws to the scaling properties of the plasma focus
Conclusions

3. Introduction (1/2)
Plasma focus: small fusion device, complements international efforts to build fusion reactor
Multi-radiation device - x-rays, particle beams and fusion neutrons
Neutrons for fusion studies
Soft XR applications include microelectronics lithography and micro-machining
Large range of device-from 0.1kJ to thousands of kJ
Experiments-dynamics, radiation, instabilities and non-linear phenomena

4. Introduction (2/2)
Scaling laws and scaling deterioration have been observed and recently explained
Plasma scaling properties have also been observed and explained.
Connection between the scaling properties and the scaling laws has yet to be made.
We now make this connection-
This new understanding paves the way for future exciting work in the Plasma Focus
Complementing the development of fusion energy

5. The Plasma Dynamics in Focus Axial Accelaration PhaseHV
30 mF, 15 kV
Axial Accelaration Phase
Inverse Pinch Phase

6. The Lee Model Code (1/8)
Realistic simulation of all gross focus properties
Couples the electrical circuit with plasma focus dynamics, thermodynamics and radiation (1984,1990)
5-phase model; axial & radial phases
Includes plasma self-absorption for SXR yield (2000)
Includes neutron yield, Yn, using a beam–target mechanism(2007)

7. The Lee Model Code (8/8) Institute for Plasma Focus Studies
Internet Workshop on Plasma Focus Numerical Experiments (IPFS-IBC1) 14 April-19 May 2008
Lee S Radiative Dense Plasma Focus Computation Package: RADPF

8. Numerical Experiments (1/2)
The Lee code is configured to work as any plasma focus:
Input:
bank parameters: L0, C0 and stray circuit resistance r0;
tube parameters: b, a and z0
operational parameters: V0 and P0 and the fill gas.
Standard practice: fit the computed total current waveform to an experimentally measured total current waveform using four model parameters :
mass swept-up factor fm;
the plasma current factor f;
for the axial phase; and
factors fmr and fcr for the radial phases.
Important information apparent from the current trace:
Axial and radial phase dynamics
Crucial energy transfer into the focus pinch

9. Numerical Experiments (2/2)
The exact time profile of the total current trace:
depends on the bank parameters, tube parameters, operational parameters., fraction of mass swept-up, fraction of sheath current in the axial and radial phases.
determines the axial and radial speeds which in turn affect the profile and magnitudes of the discharge current.
reflects the Joule heating and radiative yields.
reflects the sudden transition of the current flow from a constricted pinch to a large column flow (at the end of pinch phase).
powers all dynamic, electrodynamic, thermodynamic and radiation processes in the various phases of the plasma focus.
contains information on all the dynamic, electrodynamic, thermodynamic and radiation processes that occur in the various phases of the plasma focus.
This explains the importance attached to matching the computed current trace to the measured current trace in the procedure adopted by the Lee model code.

10. Scaling laws for neutrons from numerical experiments over a range of energies from 10kJ to 25 MJ (1/4)
To study the neutrons emitted by PF1000-like bank energies from 10kJ to 25 MJ.
1) Apply the Lee model code to fit a measured current trace of the
PF1000:
C0 = 1332 μF, V0 = 27 kV, P0 = 3.5 torr D2; b = 16 cm, a = cm or
c=1.39; z0 = 60 cm; external (or static) inductance L0= 33.5 nH and;
damping factor RESF= 1.22 (or stray resistance r0=6.1 mΩ).
2) Apply the Lee model code to the MJ machine PF1000 over a range of C0
ranging from 14 µF (8.5 kJ) to µF (24 MJ):
Voltage, V0 = 35 kV; P0 = 10 torr deuterium; RESF = 1.22; ratio c=b/a is 1.39.
For each C0, anode length z0 is varied to find the optimum z0.
For each z0, anode radius a0 is varied to get end axial speed of 10 cm/µs.

11. Scaling laws for neutrons from numerical experiments over a range of energies from 10kJ to 25 MJ (2/4)
Fitted model parameters : fm = 0.13, fc = 0.7, fmr = 0.35 and fcr=0.65.
Computed current trace agrees very well with measured trace through all the phases: axial and radial, right down to the bottom of the current dip indicating the end of the pinch phase as shown below.
PF1000:
C0 = 1332 μF; V0 = 27 kV;
P0 = 3.5 Torr D2; b = 16 cm;
a = cm; z0 = 60 cm;
L0= 33.5 nH; r0 = 6.1 mΩ or RESF=1.22.

12. Scaling laws for neutrons from numerical experiments over a range of energies from 10kJ to 25 MJ (3/4)
Voltage, V0 = 35 kV; P0 = 10 torr deuterium; RESF = 1.22; ratio c=b/a is 1.39.
Numerical experiments: C0 ranging from 14 µF(8.5 kJ) to µF (24 MJ)
For each C0, anode length z0 is varied to find the optimum z0.
For each z0, anode radius a0 is varied to get end axial speed of 10 cm/µs.
Yn scaling changes:
Yn~E02.0 at tens of kJ
Yn~E00.84 at the highest
energies (up to 25MJ)

13. Scaling laws for neutrons from numerical experiments over a range of energies from 10kJ to 25 MJ (4/4)
Scaling of Yn with Ipeak and Ipinch:
Yn=3.2x1011 Ipinch4.5
and
Yn=1.8x1010 Ipeak3.8
where Ipeak = ( )MA
and Ipinch = ( )MA.

14. Scaling laws for neon SXR from numerical experiments over a range of energies from 0.2 kJ to 1 MJ (1/4)
To study the neon SXR emitted by a modern fast bank energies from 0.2 kJ to 1 MJ.
Apply the Lee model code to a proposed modern fast plasma focus machine:
1) With optimised values:
c=b/a =1.5
V0 = 20 kV
L0= 30 nH
RESF = 0.1
Model parameters : fm=0.06, fc=0.7, fmr=0.16, fcr=0.7.
2) For C0 varying from 1 μF (0.2 kJ) to 5000 μF (1MJ):
For each C0, vary P0, z0, and a0 to find the optimum Ysxr

15. Scaling laws for neon SXR from numerical experiments over a range of energies from 0.2 kJ to 1 MJ (2/4)
Computed Total Current versus Time
For L0 = 30nH; V0 = 20 kV; C0 = 30 uF; RESF = 0.1; c=1.5
Model parameters : fm = 0.06, fc = 0.7, fmr =0.16, fcr = 0.7
Optimised a=2.285cm; b=3.43 cm and z0=5.2 cm.

16. Scaling laws for neon SXR from numerical experiments over a range of energies from 0.2 kJ to 1 MJ (3/4)
Ysxr scales as:
E01.6 at low energies in
the 0.2 to several kJ
region.
E at high energies
towards 1MJ.

17. Scaling laws for neon SXR from numerical experiments over a range of energies from 0.2 kJ to 1 MJ (4/4)
Ysxr~Ipeak3.2 (0.1–2.4 MA)
and
Ysxr~Ipinch3.6 ( MA)
Black data points with fixed parameters RESF=0.1; c=1.5; L0=30nH; V0=20 kV and model parameters fm=0.06, fc=0.7, fmr=0.16, fcr=0.7.
White data points are for specific machines with different values for the parameters :c, L0, V0 etc.

18. Summary-Scaling Laws (1/2)
The scaling laws obtained (at optimized condition) for Neutrons:
Yn~E02.0 at tens of kJ to
Yn~E00.84 at the highest energies (up to 25MJ)
Yn =3.2x1011Ipinch4.5 ( MA)
Yn=1.8x1010Ipeak ( MA)

19. Summary-Scaling Laws (2/2)
The scaling laws obtained (at optimized condition) for neon SXR:
Ysxr~E01.6 at low energies
Ysxr~E00.8 towards 1 MJ
Ysxr~Ipeak (0.1–2.4 MA) and
Ysxr~Ipinch ( MA)

20. Global scaling law, combining experimental and numerical data- Yn scaling , numerical experiments from 0.4 kJ to 25 MJ (solid line), compared to measurements compiled from publications (squares) from 0.4 kJ to 1 MJ.

21. Scaling Properties
3 kJ machine
1000 kJ machine

22. Comparing small (sub kJ) and large (thousand kJ) Plasma Focus Scaling Properties: size (energy) , current, speed and yield

23. Scaling of anode radius, current and Yn with energy E0
Peak current Ipeak increases with E0.
Anode radius ‘a’ increases with E0.
Current per cm of anode radius (ID) Ipeak /a : narrow range 160 to 210 kA/cm
SF (speed factor) (Ipeak /a)/P0.5 :
narrow range 82 to 100 (kA/cm) per Torr 0.5 D Observed Peak axial speed va : 9 to 11 cm/us.
Fusion neutron yield Yn :
106 for PF400-J to 1011 for PF1000.

24. Variation of ID SF and Yn
ID and SF are practically constant at around 180 kA/cm and 90 (kA/cm) per torr0.5 deuterium gas throughout the range of small to big devices
Yn changes over 5 orders of magnitude.

25. Comparing small (sub kJ) & large (thousand kJ) Plasma Focus Scaling Properties: size (‘a’) , T, pinch dimensions & duration

26. Focus Pinch T, dimensions & lifetime with anode radius ‘a’
Dimensions and lifetime scales as the anode radius ‘a’.
rmin/a (almost constant at )
zmax/a (almost constant at 1.5)
Pinch duration narrow range 8-14 ns/cm of ‘a’
Tpinch is measure of energy per unit mass.
Quite remarkable that this energy density varies so little (factor of 5) over such a large range of device energy (factor of 1000).

27. Comparing Deuterium & Neon (for SXR) Plasma Focus Scaling Properties: pinch dimensions & duration

28. Rule-of-thumb scaling properties, (subject to minor variations caused primarily by the variation in c=b/a) over whole range of device
Axial phase energy density (per unit mass) constant
Radial phase energy density (per unit mass) constant
Pinch radius ratio constant
Pinch length ratio constant
Pinch duration per unit anode radius constant

29. Deterioration and eventual saturation of Ipeak as capacitor energy increases

30. In numerical experiments we showed:
Yn~Ipinch4.5
Yn~Ipeak3.8
Hence saturation of Ipeak will lead to saturation of Yn.

31. What causes current scaling deterioration and eventual saturation? 1/3
The axial speed loads the discharge circuit with a dynamic resistance
The same axial speed over the range of devices means the same dynamic resistance constituting a load impedance DR0
Small PF’s : have larger generator impedance Z0=[L0/C0]^0.5 than DR0
As energy is increased by increasing C0, generator impedance Z0 drops

32. What causes current scaling deterioration and eventual saturation? 2/3
At E0 of kJ and tens of kJ the discharge circuit is dominated by Z0
Hence as E0 increases, I~C0-0.5
At the level typically of 100 kJ, Z0 has dropped to the level of DR0; circuit is now no longer dominated by Z0; and current scaling deviates from I~C0-0.5, beginning of current scaling deterioration.
At MJ levels and above, the circuit becomes dominated by DR0, current saturates

33. What causes current scaling deterioration and eventual saturation? 3/3
Analysis using the Lee model code has thus shown that the constancy of the dynamic resistance causes the current scaling deterioration resulting in the deterioration of the neutron yield and eventual saturation.
This puts the global scaling law for neutron yield on a firmer footing

34. Connecting the scaling properties with the global scaling law (1/3)
At kJ level; experimentally observedYn~E02
Ideal scaling at the highest convenient voltage V0: I~ V0 /Z0 at low energy level where Z0 dominates
leading to I~E00.5 for optimised low L0
and Yn~I04
At higher energy at around 100kJ, Z0 domination ends and current deterioration starts

35. Connecting the scaling properties with the global scaling law (2/3)
Lower current increase than the ideal leads to lower increase in anode radius ‘a’
This leads to lower increase in pinch volume and pinch duration
Which leads to lower increase in yield

36. Connecting the scaling properties with the global scaling law (3/3)
Finally at very high energies, current hardly increases anymore with further increase in energy
The anode radius should not be increased anymore; only its length should be increased
Hence pinch volume and duration also will not increase anymore.

37. Conclusion We have looked at: Neutron scaling laws with energy
Scaling properties of the Plasma Focus
The Lee model code
Neutron scaling law deterioration and saturation-the cause
and finally:
Connected the behaviour of the scaling laws to the scaling properties of the plasma focus

38. THANK YOU
Website: 1) 2) 3)

39. Papers from Lee model code
S Lee and S H Saw, “Pinch current limitation effect in plasma focus,” Appl. Phys. Lett. 92, 2008,
S Lee and S H Saw, “Neutron scaling laws from numerical experiments,” J Fusion Energy 27, 2008, pp
S Lee, P Lee, S H Saw and R S Rawat, “Numerical experiments on plasma focus pinch current limitation,” Plasma Phys. Control. Fusion 50, 2008, (8pp).
S Lee, S H Saw, P C K Lee, R S Rawat and H Schmidt, “Computing plasma focus pinch current from total current measurement,” Appl. Phys. Lett. 92 , 2008,
S Lee, “Current and neutron scaling for megajoule plasma focus machine,” Plasma Phys. Control. Fusion 50, 2008, , (14pp).
S Lee and S H Saw, “Response to “Comments on ‘Pinch current limitation effect in plasma focus’”[Appl. Phys. Lett.94, (2009)],” Appl. Phys. Leet.94, 2009,
S Lee, S H Saw, L Soto, S V Springham and S P Moo, “Numerical experiments on plasma focus neutron yield versus pressure compared with laboratory experiments,” Plasma Phys. Control. Fusion 51, 2009, (11 pp).
S H Saw, P C K Lee, R S Rawat and S Lee, “Optimizing UNU/ICTP PFF Plasma Focus for Neon Soft X-ray Operation,” accepted for publication in IEEE Trans. on Plasma Science.
Lee S, Rawat R S, Lee P and Saw S H. “Soft x-ray yield from NX2 plasma focus- correlation with plasma pinch parameters” (to be published)
S Lee, S H Saw, P Lee and R S Rawat, “Numerical experiments on plasma focus neon soft x-ray scaling”, (to be published).